Atlas › Modular forms › Weight 2 › Level 362 › 362.2.a › 362.2.a.f

Newform orbit 362.2.a.f

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 5q^{2} + 5q^{4} - q^{5} + 6q^{7} + 5q^{8} + 11q^{9} - q^{10} + 4q^{11} - 5q^{13} + 6q^{14} + 5q^{16} + 11q^{18} + q^{19} - q^{20} + 4q^{22} + 15q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{5}$ , whose minimal polynomial is

$x^{5} + 3x^{4} - 166x^{3} - 192x^{2} + 3590x + 7415$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
181	$+1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	5	0	−1	6	4	−5	0	1	15	5	−9	21	−12	4	−15	7	−24	−10	1	3	1	12	−15	0	30

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.195 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 181, 362]: [0, 1, 15, 47] (certified at ell = 2147482801)
T = [(3, 1), (5, 3)]; h of degree 16 = 1 x 16 (phi(ord eps) x dim S_k^new); 3 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(362, 2)[5]   # 362.2.a.f

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 362:2 1000